Positivity Constraints on Chiral Perturbation Theory Pion-Pion Scattering Amplitudes

Abstract

We test the positivity property of the chiral perturbation theory (ChPT) pion-pion scattering amplitudes within the Mandelstam triangle. In the one-loop approximation, $\mathcal{O}(p^4)$, the positivity constrains only the coefficients $b_3$ and $b_4$, namely one obtains that $b_4$ and the linear combination $b_3+3b_4$ are positive quantities. The two-loops approximation gives inequalities involving all the six arbitrary parameters entering ChPT amplitude, but the corrections to the one-loop approximation results are small. ChPT amplitudes pass unexpectedly well all the positivity tests giving strong support to the idea that ChPT is the good theory of the low energy pion-pion scattering.

15 pages, Latex

References in corpus (2)

• The Low Energy Expansion for Pion-Pion Scattering and Crossing Symmetry in Dispersion Relations
• Chiral two-loop pion-pion scattering parameters from crossing-symmetric constraints

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• Cornering Large-$N_c$ QCD with Positivity Bounds
• Aspects of the low-energy constants in the chiral Lagrangian for charmed mesons
• Universal Bounds for SU(3) Low Energy Constants
• NNLO Positivity Bounds on Chiral Perturbation Theory for a General Number of Flavours
• Positivity Bounds on Massive Vectors
• Determination of SU(2) Chiral Perturbation Theory low energy constants from a precise description of pion-pion scattering threshold parameters
• Positivity constraints on the low-energy constants of the chiral pion-nucleon Lagrangian